Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla

754 Pages·2019·6.936 MB·English

by  Aditya Kolachana, K. Mahesh, K. Ramasubramanian

Sources and Studies in the History of Mathematics
and Physical Sciences
Aditya Kolachana
K. Mahesh
K. Ramasubramanian Editors
Studies in
Indian Mathematics
and Astronomy
Selected Articles of Kripa Shankar
Shukla
Sources and Studies in the History
of Mathematics and Physical Sciences
Series Editors
Jed Z. Buchwald, Division of the Humanities and Social Sciences, Caltech,
Pasadena, CA, USA
Associate Editors
A. Jones, Department of Classics, Institute for the Study of the Ancient World,
New York, NY, USA
J. Lützen, Koebenhavn OE, Denmark
J. Renn, Max Planck Institute for the History of Science, Berlin, Germany
Advisory Board
C. Fraser
T. Sauer
A. Shapiro
Sources and Studies in the History of Mathematics and Physical Sciences was
inaugurated as two series in 1975 with the publication in Studies of Otto
Neugebauer’s seminal three-volume History of Ancient Mathematical Astronomy,
whichremainsthecentralhistoryofthesubject.Thispublicationwasfollowedthe
next year in Sources by Gerald Toomer’s transcription, translation (from the
Arabic), and commentary of Diocles on Burning Mirrors. The two series were
eventually amalgamated under a single editorial board led originally by Martin
Klein (d. 2009) and Gerald Toomer, respectively two of the foremost historians of
modern and ancient physical science. The goal of the joint series, as of its two
predecessors, is to publish probing histories and thorough editions of technical
developments in mathematics and physics, broadly construed. Its scope covers all
relevant work from pre-classical antiquity through the last century, ranging from
Babylonian mathematics to the scientific correspondence of H. A. Lorentz. Books
in this series will interest scholars in the history of mathematics and physics,
mathematicians, physicists, engineers, and anyone who seeks to understand the
historical underpinnings of the modern physical sciences.
More information about this series at http://www.springer.com/series/4142
Aditya Kolachana K. Mahesh
(cid:129) (cid:129)
K. Ramasubramanian
Editors
Studies in Indian
Mathematics and Astronomy
Selected Articles of Kripa Shankar Shukla
123
Editors
AdityaKolachana K.Mahesh
Indian Institute of Technology Bombay Department ofHumanities
Mumbai,Maharashtra, India andSocial Sciences
Indian Institute of Technology Bombay
K.Ramasubramanian Mumbai,Maharashtra, India
Department ofHumanities
andSocial Sciences
Indian Institute of Technology Bombay
Mumbai,Maharashtra, India
ISSN 2196-8810 ISSN 2196-8829 (electronic)
Sources andStudiesin the History of Mathematics andPhysical Sciences
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https://doi.org/10.1007/978-981-13-7326-8
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॥ प(cid:1638)सुमा(cid:1363)(cid:2248)लः ॥
K. S. Shukla
(1918–2007)
द(cid:1548)ैद(cid:1548)(cid:1011) (cid:2246)वभू(cid:2246)तम(cid:1626)(cid:2250)णतभ(cid:1024) (cid:930)ा(cid:1817)ावधेशा(cid:1707)ुदा
नैर(cid:1698)य(cid:1011)गवेषणेन मु(cid:2244)नव(cid:875) (cid:920)(cid:1699)(cid:930)काशे रतः ।
(cid:1335)ो(cid:2246)त(cid:2246)व(cid:1011)(cid:823)कृतीः (cid:2018)भूषयदयं (cid:2120)ीयानुस(cid:1701)ानतः
(cid:2246)व(cid:1637)(cid:1707)(cid:1503)लकेसर(cid:2215) (cid:2120)म(cid:2245)हमा-(cid:986)(cid:1095)ः कृपाश(cid:1216)रः ॥
Kripa Shankara Shukla, having received the treasure of the lumi-
nescence of mathematics bequeathed by [Bibhutibhusana] Datta
through Avadhesha [Narayana Singh], happily dedicating himself
to research, relentlessly worked towards the publication of [mul-
tiple] treatises like a sage. Through his investigations, he embel-
lished the writings of astronomers, and was truly a lion in the
assembly of scholars, shining (śukla) through his fame.
॥ ��समप�ण� ॥
���ाचाय�पदे �ह ल�णपुरे �व�ा�भरामः कृपा-
��ो�ो�तषशा�गूढ�वषया�लोकायचोपाहर�।
लेख���सुधी��या�सुम�नभा�स�ृ�माला�मम�
सौल�ाय बुभ�ु ुभृ�ततये द�ो वयं त�ु ये ॥
While remaining in city of Lucknow (lakṣmaṇapura) as an Ācārya,
Kripa[Shankar]Shuklawasindeedasourceofdelightforthewhole
world (viśvābhirāma), [as] he also presented the subtler aspects
of [Indian] astronomy [in a manner comprehensible] to the entire
mankind (loka). Having collected his articles resembling a bunch
of flowers, which are a source of delight to the men of wisdom
(sudhīpriya), wepresentthem, forthesakeofeasyaccessibility, as
a garland for the gratification of the swarm of bees constituted by
the [community of] knowledge-seekers (bubhutsu).
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . xix
I The Oeuvre of Kripa Shankar Shukla 1
Reminiscences of Prof. K. S. Shukla
Yukio Ohashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Dr. Kripa Shankar Shukla, veteran historian of Hindu
astronomy and mathematics
R. C. Gupta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Obituary: Kripa Shankar Shukla (1918–2007)
Yukio Ohashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Prof. Shukla’s contribution to the study of the history
of Hindu astronomy
Yukio Ohashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
The seminal contribution of K. S. Shukla to our under-
standing of Indian astronomy and mathematics
M. D. Srinivas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
II Studies in Indian Mathematics: Bhāskara I to
Nārāyaṇa Paṇḍita 71
Hindu mathematics in the 7th century as found in
Bhāskara I’s commentary on the Āryabhaṭīya (I) . . . . . 73
Hindu mathematics in the 7th century as found in
Bhāskara I’s commentary on the Āryabhaṭīya (II) . . . . 88
Hindu mathematics in the 7th century as found in
Bhāskara I’s commentary on the Āryabhaṭīya (III) . . . . 107
x Contents
Hindu mathematics in the 7th century as found in
Bhāskara I’s commentary on the Āryabhaṭīya (IV) . . . . 110
On Śrīdhara’s rational solution of Nx2+1=y2 . . . . . . . 120
Ācārya Jayadeva, the mathematician . . . . . . . . . . . . . 133
Series with fractional number of terms. . . . . . . . . . . . 153
Hindu methods for finding factors or divisors of a number 160
Magic squares in Indian mathematics . . . . . . . . . . . . 169
III Revised version of the Manuscript of the Third
Volume of Datta and Singh 187
Hindu geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Hindu trigonometry. . . . . . . . . . . . . . . . . . . . . . . . 268
Use of calculus in Hindu mathematics . . . . . . . . . . . . 345
Use of permutations and combinations in India . . . . . . 356
Magic squares in India . . . . . . . . . . . . . . . . . . . . . . 377
Use of series in India . . . . . . . . . . . . . . . . . . . . . . . 438
Surds in Hindu mathematics . . . . . . . . . . . . . . . . . . 462
Approximate values of surds in Hindu mathematics . . . 475
IV Studies in Indian Astronomy: From Vedic Period
to the Emergence of Siddhāntas 485
Astronomy in ancient and medieval India . . . . . . . . . . 487
Main characteristics and achievements of ancient Indian
astronomy in historical perspective . . . . . . . . . . . . . . 495
On three stanzas from the Pañcasiddhāntikā . . . . . . . . 510
The Pañcasiddhāntikā of Varāhamihira(1) . . . . . . . . . 517
Contents xi
The Pañcasiddhāntikā of Varāhamihira(2) . . . . . . . . . 533
Āryabhaṭa I’s astronomy with midnight day-reckoning . 548
Glimpses from the Āryabhaṭasiddhānta . . . . . . . . . . . 569
V Development of Siddhāntic Astronomy:
Some Highlights 577
Early Hindu methods in spherical astronomy . . . . . . . 579
Use of hypotenuse in the computation of the equation of
the centre under the epicyclic theory in the school of
Āryabhaṭa I ??? . . . . . . . . . . . . . . . . . . . . . . . . . . 600
Hindu astronomer Vaṭeśvara and his works . . . . . . . . . 616
The evection and the deficit of the equation of the centre
of the Moon in Hindu astronomy . . . . . . . . . . . . . . . 625
Phases of the Moon, rising and setting of planets and stars
and their conjunctions . . . . . . . . . . . . . . . . . . . . . . 646
VI Reviews and Responses 695
Vedic Mathematics: The deceptive title of Swamiji’s book 697
A note on the Rājamṛgāṅka of Bhoja published by the
Adyar Library . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
Review of Rājamṛgāṅka of Bhojarāja. . . . . . . . . . . . . 707
Review of Karaṇaratna of Devācārya
Raymond P. Mercier . . . . . . . . . . . . . . . . . . . . . . . . . 709
A note on Raymond P. Mercier’s review of “Karaṇaratna
of Devācārya” . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
The yuga of the Yavanajātaka: David Pingree’s text and
translation reviewed . . . . . . . . . . . . . . . . . . . . . . . 719
Review of Vaṭeśvarasiddhānta and Gola of Vaṭeśvara
David Pingree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
Appendix . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . .734